Table Of ContentOptical conductivity of a 2DEG with anisotropic Rashba interaction at the interface
of LaAlO /SrTiO
3 3
Alestin Mawrie and Tarun Kanti Ghosh
Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India
(Dated: May 12, 2016)
Westudyopticalconductivityofatwo-dimensionalelectron gaswithanisotropic k-cubicRashba
6
1 spin-orbit interaction formed at the LaAlO3/SrTiO3 interface. The anisotropic spin splitting en-
ergygivesrisetodifferentfeaturesoftheopticalconductivityincomparisontotheisotropick-cubic
0
Rashbaspin-orbitinteraction. Forlargecarrierdensityandstrongspin-orbitcouplings,thedensity
2
dependence of Drude weight deviates from the linear behavior. The charge and optical conduc-
y tivities remain isotropic despite anisotropic nature of the Fermi contours. An infinitesimally small
a photon energy would suffice to initiate inter-band optical transitions due to degeneracy along cer-
M
taindirectionsinmomentumspace. Theopticalconductivityshowsasinglepeakatagivenphoton
energy depending on the system parameters and then falls off to zero at higher photon energy.
1 These features are lacking for systems with isotropic k-cubic Rashba spin-orbit coupling. These
1 striking features can be used to extract the information about nature of the spin-orbit interaction
experimentally and illuminate some light on theorbital origin of thetwo-dimensional electron gas.
]
l
l PACSnumbers: 78.67.-n,72.20.-i,71.70.Ej
a
h
-
s I. INTRODUCTION and are localized at the interface due to impurities and
e electron-phonon coupling18, whereas the electrons as-
m
Spin-orbitinteraction1,2 (SOI)playsanimportantrole sociated with the dxz and dyz orbitals18 are itinerant
. and contribute to transport. One of the major con-
t in understanding physical properties of different materi-
a cerns is to understand the nature of the SOI of the
m als as it lifts the spin degeneracy due to the absence of
charge carriers at the oxide interface. In Refs.17,19, a
eitherthestructureinversionsymmetryorthetimerever-
- k-linear Rashba SOI for d orbital and an isotropic
d salsymmetry. Ingeneral,therearetwodifferenttypesof xy
n symmetry dependent SOI, Rashba3,4 and Dresselhaus5 k-cubic for dxz and dyz orbitals were proposed. The
o SOIs, in various condensed matter systems. In two- magneto-transport measurement of 2DEG at the oxide
c dimensional electron gas (2DEG) formed at the III-V interfacehasindicatedtheexistenceofk-cubicRSOIand
[ semiconductorheterostructures6 andinvarioustopologi- ismodeledusingtheisotropick-cubicSOI,HRiso10,20. On
2 calinsulatingsystems7,the RashbaSOI(RSOI)islinear the other hand, the first-principle calculations suggested
1v iαnimsothmeesnttruemngathndofofRtShOeIfo,rσm±H=Rσ=x iαiσky−σw+it+hhσ.cx.,awndheσrye aannidsodtyrzopoicrbnitoanl-sp2a1.rabTowlioc srpecinen-stplpitolbarraiznacthioesn-fdoerptehneddexnzt
9 ± ARPES revealed non-isotropic Fermi contours of the
5 are the Pauli’s spin matrices and k± =kx±iky with kx 2DEG at the oxide interface22. Very recent theoretical
andk thecomponentsofthewavevectork. Besides,the
0 y study23 predicted that these orbitals are characterized
0 Rashba SOI in two-dimensional hole gas formed at the
. interface of p-type GaAs/AlGaAs heterostructures8,9, by k-cubic but anisotropic Rashba spin-orbit interaction
01 2DEG on the surface of SrTiO3 single crystals10 and wsphinossepfloitrtminigsegniveerngybyanHdRaFnier=mαi(ckox2n−tokuy2rs)(bke×coσm)e·zˆh.igThhlye
in 2D hole gas formed in a strained Ge/SiGe quan-
6
1 tum well11 is cubic in momentum and is of the form anisotropicasaresultofthisanisotropicSOI.Inthispa-
v: HRiso = iαk−3σ+ + h.c. The spin splitting energy due ptheerawneiswotirllopreicfeRrStOhiIs23asenaanbisleostrtoopeixcpRlaSiOnIt.heTehxispeforirmmeno-f
to this RSOI is always isotropic and hereafter we will
Xi mention this as isotropic cubic RSOI. talobservationsoftheanisotropicspinsusceptibility24,25
successfully. Itisalsoshown23thattheanisotropicRSOI
r An extremely high mobility 2DEG was discovered
a leads to different behavior of the spin Hall conductivity,
at the interface of the complex oxides LaAlO and
3 in comparisonto the isotropic k-cubic RSOI.
SrTiO 12–14. Themobilityatoxideinterfacesisrelatively
3
lessthanthatofIII-Vsemiconductorheterostructures16. The spectroscopicmeasurementofthe absorptivepart
Unlike the conventionalIII-V semiconductorheterojunc- oftheopticalconductivitycanprobethespin-splitenergy
tions,the 2DEGsatLaAlO /SrTiO interfacesarechar- levels. Theoretical studies of the optical conductivity of
3 3
acterized by very strong spin-orbit interaction, high car- variouschargedsystemswithanisotropick-cubicRashba
rier densities, higher effective mass15. The LAO/STO SOI have been carried out26–29,31. It is shown that the
interface structure now has a broken structure inver- opticaltransitiontakesplaceforacertainrangeofphoton
sion symmetry as a result of the confinement along the energy depending on the carrier density and spin-orbit
axis normal to the interface, which leads to the lifting coupling constant. At zero temperature, it takes a box-
of the spin-degeneracy of the six t orbitals in STO17. like function and its value is σiso = 3e2/(16~), indepen-
2g xx
Moreover, the d orbitals are confined in the x-y plane dent of carrier density and spin-orbit coupling strength.
xy
2
In this paper we study the Drude weight and opti- Here Ω is the surface area of the two-dimensional sys-
cal conductivities of the 2DEG with anisotropic k-cubic tem, λ = denotes the spin-split branches and ηk =
RSOI formed at the oxide interface and compare our re- cos2θ/ cos±2θ with θ = tan−1(k /k ) measures the
y x
| |
sults with that of the isotropick-cubic RSOI.Firstly, we anisotropy of the spectrum. The magnitude of the
present the characteristics of the zero-frequency Drude anisotropic spin-splitting energy is E (k) = E (k)
g +
weight as a function of the charge density and strength E−(k) = 2αk3 cos2θ . The spin splitting en|ergy van−-
| | |
oftheanisotropicRSOI.WefindthattheDrudeweightis ishes at θ = (2p + 1)π/4 with p = 0,1,2,3. On the
strongly modified due to the presence of the anisotropic other hand, the maximum spin-splitting (Emax = 2αk3)
g
k-cubic SOI. It deviates from the linear density depen- occurs at θ = pπ/2. To allow only the bound states,
dence for large carrier density and for strong spin-orbit the wave-vectork should have an upper cut-off given by
coupling. The Drude weight decreases with the increase k (π/4) = ~2/(3m∗α) which corresponds to the cut-off
c
of the strength of RSOI. Secondly, we find that an in- energy E =αk3/2.
c c
finitesimallysmallphotonenergywouldinitiatetheinter- The spin texture on the k -k plane can be obtained
x y
bandopticaltransition. Thisisduetothevanishingspin- from the average values of spin vector (in units of 3~/2)
splitting energy along certain directions in the momen- Pλ(k)= σ λ =ληkθˆ,whereθˆ= xˆsinθ+yˆcosθ is the
tum space. There is a single peak in the optical conduc- unit polahr viector. The electron sp−in lies in the k plane
tivity and its value depends on the electron density and and always locked at right angles to its momentum.
strength of the anisotropic RSOI. Moreover, the charge The Berry connection30 is defined as Ak = i φλk k
and optical conductivities are isotropic despite the fact φλ , where φλ is the spinor part of the wave fhunc|t▽ion
k k
that the RSOI is anisotropic. In conventional2DEG the ψ| kλ(ir). TheBerryconnectionforthissystemyieldsAk =
vanHovesingularitieslargelyaffectsthevariousphysical θˆ/(2k). Using the expression of the Berry phase30 γ =
ptyrpopeekritnieksinlikpehtortaonluspmoirnte3s2c,ecnhcae3r4a,ctdeirluotfe-pmlaasgmneotnics3s3e,mNi-- H−γisAok=·d3kπ,fworeigseottrγoapniic=cu−bπicfRorSaOnIi3s1o.tropiccase,whereas
conductor properties35 etc. Here as well, the van Hove
singularities drastically affects the optical conductivity,
thereby responsible for the single peak observed in it.
The van Hove singularities are of the same M1 type. 1.15 D−
These features can be used to find out the nature of the
RSOI experimentally.
1.1
D−
This paper is organized as follows. In section II, we
describebasicpropertiesofthe2DEGwithanisotropick-
cubicspin-orbitinteraction. InsectionIII,wepresentthe )1.05
D0
analyticalandnumericalresultsoftheDrudeweightand
(
the optical conductivity. The summary and conclusions λ
D 1
of this paper are presented in section IV.
0.95
II. DESCRIPTION OF THE PHYSICAL D+
SYSTEM
0.9 D+
0 2 4 6 8 10 12
The effective Hamiltonian of the electron in d and
xz E (meV)
d orbitals at the interface of LAO/STO is given by23
yz
~2k2 FIG. 1: (color online) Plots showing the density of states in
H = 2m∗ +α(kx2 −ky2)(k×σ)·zˆ, (1) teVhenumn3its(soofliDd0blfaocrkt)waonddiffαe=ren0t.0v0a6lueeVsnomf α3.(dHoetrteedαr=ed0)..004
where m∗ is the effective mass of the electron, α is the
strength of the anisotropic RSOI and σ = σ xˆ +σ yˆ.
x y
In order to obtain two anisotropic Fermi contours
TheaboveHamiltonianisvalidwithinthe narrowregion
kλ(θ), we need to calculate density of states (DOS) and
aroundthe Γ point. The anisotropic dispersion relations f
Fermi energy E . The density of states of the spin-split
and the corresponding eigenfunctions are given by f
energy branches are given by
E (k)= ~2k2 +λαk3 cos2θ (2) d2k
λ 2m∗ | | Dλ(E) = Z (2π)2δ(E−Eλ(k))
and ψkλ(r)=eik·rφλk(r)/√Ω with the spinor = D0 2π kEλ(θ)dθ ,
1 1 2π Z0 |kEλ(θ)+λ6παD0(kEλ(θ))2|cos2θ||
φλk(r)= √2(cid:18)ληkieiθ(cid:19). (3) whereD0 =2πm∗/h2andkEλ(θ)beingthesolutionofthe
3
equation (~kλ)2/2m∗ +αλ(kλ)3 cos2θ E = 0. The of the frequency of the AC electric field with vanishing
E E | |−
density of states is obtained numerically and their char- momentumq 0. Thevanishingmomentumoftheelec-
→
acteristicsforthe twobranchesareshowninFig. 1. The tric field forces the charge carriers to make a transition
DOS of the anisotropic spin-split levels varies asymmet- from λ = 1 branch to λ = +1 branch such that the
−
rically with respect to D . For fixed electron density n momentum is conserved.
0 e
andα,theFermienergy(E )isobtainedfromtheconser- Drude weight: The semi-classical expression for the
f
vation of electron number n = Ef D (E)dE. The Drude weight at low temperature is given by36
e 0 λ λ
variations of the Fermi energy wRith nPe and α are shown d2k
in Fig. 2. The Fermi energy increases with the increase Dw =πe2 Z (2π)2 hvˆxi2λδ(Eλ(k)−Ef). (4)
of the carrier density. On the other hand, the Fermi Xλ
energy decreases with the increase of the spin-orbit cou- Herevˆ is the x-componentofthe velocityoperator. Us-
x
pling strength. The Fermi wave vectors kfλ(θ) can be ingtheHeisenberg’sequationofmotion,i~r˙ =[r,H],the
obtained numerically from the solutions of the equation x- and y-components of the velocity operator are given
~2k2/2m∗+λαk3 cos2θ Ef =0. The Fermi contours by
| |−
are depicted in Fig. 5 (color: black).
~k α
vˆ = xI+ (3k2 k2)σ 2k k σ (5)
x m∗ ~ x− y y − x y x
(cid:2) (cid:3)
18 α=0.004eVnm3 9.5 and
α=0.008eVnm3 ~k α
16 α=0.012eVnm3 9 ne=4.0×1016m−2 vˆy = m∗yI− ~ (3ky2−kx2)σx−2kxkyσy . (6)
14 ne=3.5×1016m−2 (cid:2) (cid:3)
) 8.5 ne=3.0×1016m−2 For the system with anisotropic cubic RSOI, the cal-
V12 culation of the Drude conductivity yields
e
m
E(f180 8 Dwani = (cid:16)2πe~(cid:17)2Xλ Z02πm∗[vfλ(θ)]2Bλ(θ)dθ, (7)
7.5
6 where vλ(θ)=~kλ(θ)/m∗ and
f f
7
4 [cosθ+ληkαVλ(θ)(5cosθ+cos3θ)/2]2
B (θ)= f (8)
λ 1+λ3αVλ(θ)
2 4 6 8 0.005 0.01 0.015 0.02 f
ne (×1016m−2) α(eVnm3) with Vλ(θ)=(m∗/~2)kλ(θ).
f f
For carrying out the numerical calculation, we adopt
FIG.2: (color online)Plots oftheFermienergyvsne andα. the following parameters used in Refs.19,23: ne = 3.5
Left panel: plots of the Fermi energy vs density for different 1016 m−2 and m∗/m = 1, where m is the bare mas×s
0 0
values of α. Right panel: plots of the Fermi energy vs α for
of the electron. In Fig. 3, the variations of the Drude
different values of carrier density ne. weight with the carrier density and with the strength of
the Rashba spin-orbit interactions are shown. The plots
of the Drude weight vs carrier density for three different
values of α are shown in the left panels of Fig. 3. The
III. DRUDE WEIGHT AND OPTICAL analytical expression of Dwiso obtained in Ref.31 clearly
CONDUCTIVITY shows the deviation from the linear density dependence.
Because of the small value of α considered here, the de-
viation is not visible in this figure. On the other hand,
The complex charge conductivity for a two-level sys-
the Drude weight vs α for three different values of car-
tem of charge carriers in presence of a sinusoidal elec-
rierdensityareplottedintherightpanelsofFig. 3. The
tric field (E(ω) xˆE eiωt) can be written as Σ (ω) =
∼ 0 xx Drudeweightdecreaseswiththeincreaseofαbutthede-
σ (ω)+σ (ω), where σ (ω) is the intra-band induced
D xx D creasingnature of D for the two different cases is quite
dynamicDrudeconductivityandσ (ω)istheinter-band w
xx different. Thisimportantfeaturewouldhelptoknowthe
induced complex optical conductivity.
nature of the RSOI.
The absorptive part of the conductivity can be ob-
Optical Conductivity: The generalized Kubo for-
tained by taking the real part of Σ (ω) and is given
xx mula of the optical conductivity in terms of the Matsub-
by
ara Green’s function is given by36
Re[Σxx(ω)]=Dwδ(ω)+Re[σxx(ω)]. e2T 1
σ (ω)= d2k
Here, D is known as the Drude weight measuring the µν − iω (2π)2 Z
w
aDnrdudReec[oσnxdxu(ωct)i]viistyth(σedop=tiτcDalwc/oπn)dufocrtiaviDtyCaesleactfurinccfitieoldn × Xl TrhvˆµGˆ(k,ωl)vˆνGˆ(k,ωs+ωl)iiωs→ω+iδ. (9)
4
× 10−3 optical conductivity at T =0 is given by
1
1 e2 2π
D)α (D)n Re[σxx(ω)]= 24hZ0 dθsin2θhΘ(µ+)−Θ(µ−)i,(12)
aniD(w aniDw wµhewreithΘ(kx)istkhe(θu)n=its(t~eωp/f2uαncctoiosn2θan)1d/3µ.±T=hiEs±in(tkeωg)r−al
0.8 0 ω w
0.2 ≡ | |
cannotbe solvedanalytically due to θ dependence ofk .
ω
0× 10−3 2 4 6 0 0.01 0.02
1 2
ε
−
1 )
) n
α D
D ( 1.5
(
o
isoDw isDw
ε
0.2 0.85 π/ 1 +
θ
0 2 4 6 0 0.01 0.02
n (×1016 m−2) α (eVnm3)
e
0.5
FIG. 3: (color online) Left panels: plots of the Drudeweight
Dw (in units of Dα = πe2/m∗lα2) vs ne for α = 0.006 eV a
nm3 (solid: blue), α = 0.008 eV nm3 (dotted-dashed: red) 0
fRaonirgdhnαtep==an30e.0l:112p0l1eo6Vtsmno−mf2D3(w(ddaa(sishnheedud:n:ibtbslalaocckfk)D),.nnHe=e=rπe3e.l25αne=/1mm0∗1∗6)αvm/s~−2α2. D(ω) 0.3 0.02.12
(doted-dash×ed: red) and ne =4 1016 m−2 (solid×: blue). × 0.2
× k0 0.19
α 0.2 0.25 0.3
2 0.1
π
4 b
H1)eπrTe,aµn,dνω=l =x,2yl,πTT baereintghethfeertmemiopneicraatnudreb,oωsson=ic(M2sa+t- 4h) 10 x 10−3 1 h)
subarafrequencieswithsandl areintegers,respectively. /2 5 24
The matrix Green’s function associated with the 2e 2/
Hamiltonian given by Eq. (1) is ω)]( 0.5 0.5 ω)](e
G(k,ωn)= 12Xλ hI+Pλ(k)·σiGλ0(k,ωn). (10) σaniRe[(xx 00 0.05 0.1 c σisoRe[(xx
0 0
Here I is a 2×2 unit matrix and Gλ0(k,ωn)=1/(i~ωn+ 0 0.2 0.4 0.6 0.8
µ0 Eλ(k)) with µ0 being the chemical potential. It in- ¯hω (meV)
−
dicatesthattheopticalspectralweightisdirectlyrelated
to the local spin texture P (k).
λ FIG. 4: (color online): Top panel: Plots of ǫ±(θ) vs θ. Mid-
Substituting Eqs. (5) and (10) into Eq. (9), the xx- dle panel: plots of the joint density of states vs ~ω with
component of the longitudinal conductivity reduces to
k0 = √2πne. Bottom panel: the real part of the optical
e2 ∞ 2π conductivityas a function of photon energy ~ω.
σ (ω) = α2k5 cos22θsin2θdkdθ
xx − i(2π~)2ω Z Z
0 0
f(E−) f(E+) On the other hand, in isotropic cubic Rashba SOI the
× h~ω+iδ −E++E− +(E− ↔E+)i, (11) closed form expression of the absorptive part of the op-
− tical conductivity at T =0 K is given by31
where f(E)=[e(E−µ0)β+1]−1 is the Fermi-Dirac distri-
buWtioenhfauvnectcioanrrwieidthoβut=th1e/(ksaBmTe).calculation for other Re[σxisxo(ω)]= 136e~2 Θ(µ˜+)−Θ(µ˜−) , (13)
components of the conductivity tensor σ (ω). We find (cid:2) (cid:3)
µν
that σyy(ω) = σxx(ω) and σxy(ω) = σyx(ω) = 0. Hence where µ˜± = E±(k˜ω) µ0 with k˜ω = (~ω/2α)1/3 =
−
theopticalconductivityremainsisotropicdespitethefact k (θ = (2p + 1)π/2). It leads to featureless opti-
w
that the Fermi contours are anisotropic. cal conductivity which has box shape with the height
Using the fact that ω > 0 and after performing the σiso =3e2/(16~)whichisindependentofthecarrierden-
xx
k integral, the expression for the absorptive part of the sity and α. Note that simultaneous presence of isotropic
5
Rashba and Dresselhaus SOI leads to anisotropic Fermi optical conductivity from other two bands having k- lin-
contours, in turns produces interesting optical features. ear SOI is ruled out since they occur at ~ω much larger
WhereasanisotropicRSOIalonegivesrisetoanisotropic than ǫ−(pπ/2) 0.9 meV.
≈
Fermi contours and provides distinct optical features. The overall behavior of the optical spectra can be un-
derstood from the joint density of states which is given
as
d2k
D(ω)=Z (2π)2[f(E+(k))−f(E−(k))]δ(Eg(k)−~ω).
1.4 1.1
1
1.2
)
h
4 0.9
2 1
/
2e 0.8
(0.8
k
a 0.7
pe0.6
σ
0.6
0.4
FIG. 5: (color online) Plots of the Fermi contours k+(θ), 0.5
f 0.2
kf−(θ), the constant-energy difference curves C1: Eg(k)= ǫ1 0 0.005 0.01 1 2 3 4
and C2: Eg(k)=ǫ2. α (eVnm3) ne (×1016m−2)
Here we shall present how the anisotropic RSOI alone FIG.6: (coloronline)Leftpanel: Plotsofσpeak vsαforfixed
gives rise to some unique features of the optical conduc- values of ne = 4.0 1016 m−2 (dashed) and ne = 3.0 1016
tivity. We first evaluate Re[σxx(ω)] numerically using m−2 (solid). Right×panel: Plots of σpeak vs ne for di×fferent
the parameters α = 0.004 eV nm3, n = 3.5 1016 m−2 values of α = 0.004 eV nm3 (solid) and α = 0.006 eV nm3
e
and m∗/m = 1 as used in Refs.19,23 and sh×own in the (dashed).
0
lower panel of Fig. 4. For comparison with the isotropic
case, we plot Re[σiso(ω)] which appears as the rectan-
xx
It can be reformulated as
gular box on the right side of the lower panel of Fig. 4.
We depict ǫ±(θ) = 2α[kf±(θ)]3|cos(2θ)| in the top panel 1 dC[f(E+(kω)) f(E−(kω))]
ofFig. 4. Thecontributiontoopticalconductivityarises D(ω)= − . (14)
from the shaded angular region. The optical transitions (2π)2 ZC |∂kEg(k)|Eg=~ω
from λ = 1 to λ = +1 occur when the photon en-
ergy satisfie−s the inequality 0 < ~ω < ǫ−(θ). One can HdeenrseitCy oisftshtaetelisnevsel~eωmeisntplaoltotnegdtihnetchoenmtoiudrd.leTphaenjeolinotf
see that an infinitesimally small photon energy can ini-
Fig. 4. The location of the single peak and the region
tiate the optical transition, in complete contrast to the
of zero optical conductivity are nicely described by the
isotropic SOI case. This is due to the presence of the
joint density of states. It can be seen from Eq. (14)
degenerate lines θ =(2p+1)π/4. There is a single peak
that any peak may arise whenever ∂ E (k) attains a
of the Re [σxx(ω)] at ~ω = ǫ+(pπ/2) = 2α[kf+(pπ/2)]3 minimum value. For the present pro|bklemg, th|e singular
and the optical conductivity becomes zero when ~ω points are at k =(k,pπ/4). The single peak appears at
− 3 ≥ s
ǫ−(pπ/2) = 2α[kf (pπ/2)] . For better understanding ǫω = ǫ+(pπ/2) in the joint density of states corresponds
of these features, we plot the constant energy-difference tothe wellknownvanHovesingularity. The asymmetric
curves E (k) = ǫ for ǫ = ǫ (pπ/2)= ǫ (C : dashed) spin-splitting at the Fermi contours along the k =k =
g ω ω + 1 1 y x
and ǫω = ǫ−(pπ/2) = ǫ2 (C2: solid) in Fig. 5. The area 0 lines is the reason for the appearance of the peak at
intercept by the curves C with i = 1,2 and the Fermi ǫ (pπ/2).
i +
contours (kλ) are responsible for the k-selective optical There are three different types of the singularity37
f
transitions as shown in Fig. 5. It should be noted that depending on the nature of change of the energy gap
the anisotropic k-cubic band is well separated from the around the singular points k . Using the Taylor series
s
other two bands with k-linear SOI for the parameters expansion of E (k) around k as E (k) = E (k ) +
g s g g s
used in Refs.19,23. As a result, the contribution to the a (p)(k k )2 with the expansion coefficients
µ µ µ − sµ
P
6
α are shown in Fig. 6. It strongly depends on the Fermi
0.8
energy. Wealsodefineawidth∆=ǫ−(pπ/2) ǫ+(pπ/2),
−
2 thedifferencebetweenpeakpositionandthepositionbe-
yondwhich σ (ω) vanishes. Its variationwith α as well
0.6 xx
asn areshowninFig. 7. Itshowsthat∆increaseswith
e
1.5 the increase of n as well as α.
) e
V
e
m0.4
( 1
∆ IV. SUMMARY AND CONCLUSION
0.2
0.5 WehavestudiedtheDrudeweightandopticalconduc-
tivity for 2DEG with k-cubic anisotropic RSOI at the
oxide interface. We have presented the variation of the
0 0
0 0.005 0.010 5 10 zero-frequency Drude weight with the carrier density as
α (eVnm3) n ×1016 m−2) well as the strength of the anisotropic spin-orbit cou-
e
pling. For anisotropic RSOI, the Drude weight deviates
from the linear density dependence. It is indicated that
the spectral weight is directly related to the local spin
FIG. 7: (color online) Left panel: Plots of ∆ vs α for fixed texture in momentum space. We found that the charge
values of ne =3.5 1016 m−2 (dashed) and ne = 3.0 1016 and optical conductivities remain isotropic although the
m−2(solid). Right×panel: Plotsof∆vsne fordifferent×values Fermi contours are anisotropic. It is found that an in-
of α=0.004 eV nm3 (solid) andα=0.006 eVnm3 (dashed).
finitesimally small photon energy can trigger inter-band
opticalconductivity. Thisisduetothefactthatthespin-
splittingenergyvanishesalongthecertaindirectionsink
2a (p) = ∂2Eg(k) . The co-efficients a are as follows space. We found a single peak in the optical conductiv-
µ ∂kµ2 (cid:12)ks µ ity whose value depends on the Fermi energy. We have
a (p)=αk[5+7((cid:12) 1)p]anda (p)=αk[5 7( 1)p]. The shownthatthevanHovesingularitiesresponsibleforthe
x (cid:12)− y − −
signofthe coefficients will determine the type of classifi- single peak in the optical conductivity are of the same
cationofthe varioussingularpoints. One caneasilyfind M type. The different features of the conductivity can
1
that the signs of a and a at different singular points determinethe informationofthe natureofthe spin-orbit
x y
are ( 1)p and ( 1)p+1, respectively. Therefore, every interactionexperimentallyandwouldhelpinunderstand-
− −
singularities are all of the same class i.e. M type. ingthe orbitaloriginofthe two-dimensionalelectrongas
1
The variations of the peak height (σ ) with n and at the oxide interface.
peak e
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